1. Neuronal Circuits CSCI 2323-1

2. Neurons http://stepitup2007.org/article.ph p?list=class&class=20&offset=40

3. Neuron Schematic http://faculty.etsu.edu /currie/excitation.htm

4. Action Potential http://openwetware.org/wiki/BIO254:AP

5. Circuits http://www.physics247.com/physic s-tutorial/parallel-circuits.shtml http://en.wikipedia.org/wiki /Parallel_circuit

6. Definitions Resistance: the opposition to the passage of a steady electric current by a material. Resistance: the opposition to the passage of a steady electric current by a material. Capacitance: the storage of energy due to separation of charge. Capacitance: the storage of energy due to separation of charge. Current: flow of electric charge. Current: flow of electric charge. Potential: the energy released in the transfer of a unit quantity of electricity from one point to the other. Potential: the energy released in the transfer of a unit quantity of electricity from one point to the other. All definitions provided by Merriam-Webster Dictionary

7. Abbreviations to be used R: Resistance R: Resistance C: Capacitance C: Capacitance I: Current I: Current V: Potential V: Potential g: conductance g: conductance m : membrane m : membrane l : longitudinal (across the membrane) ‏ l : longitudinal (across the membrane) ‏ i : ion (in this case, Na +, K + & Cl -) i : ion (in this case, Na +, K + & Cl -) a: radius a: radius

8. Modeling the neuron as a circuit

9. Assumptions All ions contribute to V in a similar way. All ions contribute to V in a similar way. Only energy cost to the neuron is the separation of charge, without distinguishing between the types of ions. Ex. Na +/ K + ATPase. Only energy cost to the neuron is the separation of charge, without distinguishing between the types of ions. Ex. Na +/ K + ATPase. Kirchoff’s First Law: Net charge can not pile up inside the individual circuit elements. Thus, the volume of ions flowing = volume of ions flowing out. Kirchoff’s First Law: Net charge can not pile up inside the individual circuit elements. Thus, the volume of ions flowing = volume of ions flowing out. Cytosol is very resistant to the movement of electric charge. Cytosol is very resistant to the movement of electric charge. [ ] and charge differences are only seen in a small region around the membrane. [ ] and charge differences are only seen in a small region around the membrane.

10. The membrane

11. The Nernst Relation V i Nernst =(-K B T/ze)*ln([in]/[out]) ‏ V i Nernst =(-K B T/ze)*ln([in]/[out]) ‏ This equation tells us something useful: the electric potential of our circuit with respect to an ion. This equation tells us something useful: the electric potential of our circuit with respect to an ion. We will see this equation A LOT!!!! We will see this equation A LOT!!!!

12. Resting Membrane Potential Donnan equilibrium helps us creates the resting membrane potential (~72 mV). Created because ions have concentration differences across the membrane. Donnan equilibrium helps us creates the resting membrane potential (~72 mV). Created because ions have concentration differences across the membrane. ION PUMPS!!!! ION PUMPS!!!! Resting membrane potential is a steady state, but not an equilibrium. Resting membrane potential is a steady state, but not an equilibrium. We can assume ∆V=V Nernst for all ions present. Thus: we can get this equation: We can assume ∆V=V Nernst for all ions present. Thus: we can get this equation: [ ] out, Na+ /[ ] in, Na+ = [ ] out, K+ /[ ] in, K+ = [ ] out, Na+ /[ ] in, Na+ = [ ] out, K+ /[ ] in, K+ = [ ] out, Cl- /[ ] in, Cl- All of this is equal to ∆V. All of this is equal to ∆V.

13. Ion flux ∆V=V Nernst is boring. ∆V=V Nernst is boring. j q,i is much more interesting, since it tells us more about the flux of ions. j q,i is much more interesting, since it tells us more about the flux of ions. Ions crossing the membranes lead to cool stuff happening (Action Potentials!). Ions crossing the membranes lead to cool stuff happening (Action Potentials!).

14. Ohm’s Law Ohm’s Law: ∆V=I i R i +V i Nernst Ohm’s Law: ∆V=I i R i +V i Nernst ∆V=Vin-Vout ∆V=Vin-Vout Potential difference across the membrane Potential difference across the membrane I i =j q,i A I i =j q,i A Current generated by ions crossing the membrane Current generated by ions crossing the membrane R i =1/g i A R i =1/g i A Resistance of the membrane to each ion. Resistance of the membrane to each ion.

15. Capacitors As before, the membrane is multi-purpose. As before, the membrane is multi-purpose. Net charge across the membrane is zero. Thus, we can’t have charge imbalance (DONNAN EQUILIBRIUM). Net charge across the membrane is zero. Thus, we can’t have charge imbalance (DONNAN EQUILIBRIUM). However, charge can accumulate in a very small region around the membrane. This allows for charge to flow toward the membrane. However, charge can accumulate in a very small region around the membrane. This allows for charge to flow toward the membrane. q=C(∆V)  d(∆V)/dt=I/C q=C(∆V)  d(∆V)/dt=I/C

16. GOOD!!! We have been able to successfully create several of the parameters modeling the neuronal circuit, including the potential difference across the membrane, the capacitance. All of this has been thanks to the innate physiology! We have been able to successfully create several of the parameters modeling the neuronal circuit, including the potential difference across the membrane, the capacitance. All of this has been thanks to the innate physiology!

17. BUT NOT GOOD ENOUGH!!! Our current model fails to accurately model a real axon. Our model assumes that the axon is small (not always the case), or a large membrane maintained at a potential that is uniform across its length. Real axons don’t work like this, since only small parts of the axon are actually propagating the action potential. Our current model fails to accurately model a real axon. Our model assumes that the axon is small (not always the case), or a large membrane maintained at a potential that is uniform across its length. Real axons don’t work like this, since only small parts of the axon are actually propagating the action potential. Some equations just don’t cut it. Some equations just don’t cut it. So, we need a little more advanced formula to model what is really going on. So, we need a little more advanced formula to model what is really going on.